What is the integral of e^{-0.5x^2} ?

The integral of e^{-0.5x^2} is 1.25331…\erf(sqrt(0.5)x)+C

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integral of e^{-0.5x^2}

\int e^{- 0.5 x^{2}} d x

1.25331 \dots erf (0.5 x) + C

Solution steps

\int e^{- 0.5 x^{2}} d x

= \int e^{- (0.5 x)^{2}} d x

Apply u-substitution

= \int e^{- u^{2}} \frac{1}{0.70710 \dots} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{0.70710 \dots} \cdot \int e^{- u^{2}} d u

Simplify

= 1.41421 \dots \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

\int e^{- u^{2}} d u = \frac{π}{2} erf (u)

= 1.41421 \dots \cdot \frac{π}{2} erf (u)

Substitute back

u = 0.5 x

= 1.41421 \dots \cdot \frac{π}{2} erf (0.5 x)

Simplify

1.41421 \dots \cdot \frac{π}{2} erf (0.5 x) : 1.25331 \dots erf (0.5 x)

= 1.25331 \dots erf (0.5 x)

Add a constant to the solution

= 1.25331 \dots erf (0.5 x) + C

What is the integral of e^{-0.5x^2} ?
The integral of e^{-0.5x^2} is 1.25331…\erf(sqrt(0.5)x)+C